Crossposted on MathOverflow.
In this post, an order-preserving self-map of a poset $X$ will be called an endomorphism of $X$, and such an endomorphism $f$ will be called rigid if the only automorphism of $X$ which commutes with $f$ is the identity of $X$.
The question is in the title:
Does every finite poset have a rigid endomorphism?
Every poset of cardinality at most $9$ has a rigid endomorphism.
I wrote a proof of this statement in a separate text. Since links tend to break over time I am including several links to this text:
pdf file --- tex file --- Overleaf --- Google Drive --- Mediafire.
In the first version of this question I put the proof in the post itself. But I realized that there was a mistake, and that the post was too long. So I rewrote the proof (hoping that it is correct now), and added links to the new version.